Giant spin ensembles in waveguide magnonics

The dipole approximation is usually employed to describe light-matter interactions under ordinary conditions. With the development of artificial atomic systems, ‘giant atom’ physics is possible, where the scale of atoms is comparable to or even greater than the wavelength of the light they interact with, and the dipole approximation is no longer valid. It reveals interesting physics impossible in small atoms and may offer useful applications. Here, we experimentally demonstrate the giant spin ensemble (GSE), where a ferromagnetic spin ensemble interacts twice with the meandering waveguide, and the coupling strength between them can be continuously tuned from finite (coupled) to zero (decoupled) by varying the frequency. In the nested configuration, we investigate the collective behavior of two GSEs and find extraordinary phenomena that cannot be observed in conventional systems. Our experiment offers a new platform for ‘giant atom’ physics.

The experiments performed by the authors demonstrate thoroughly and convincingly how they can harness the interference effects due to the multiple coupling points of giant atoms. By tuning the resonance frequency of a spin ensemble using a bias magnetic field, the phase acquired when travelling between coupling points changes. This leads to either constructive or destructive interference, resulting in either high or low relaxation rate from the atom into the waveguide. The authors further demonstrate what happens when two giant atoms in a "nested" configuration (the two coupling points of one atom placed in between the two coupling points of the other atom) have their frequencies tuned. They find points where the coupling between the two atoms is purely dissipative and where it is purely coherent. In the latter case, the individual relaxation rate of one of the giant atoms also goes to zero.
The paper is quite well written, the figures are very nice, and the main text together with the supplementary material contains all necessary information to understand the experiments. The authors have demonstrated a new experimental platform to investigate the physics of giant atoms, which, as stated in the outlook, may offer some interesting possiblities compared to previous platforms with superconducting qubits. In light of this, I believe that the paper eventually can become suitable for publication in Nature Communications. However, before that, the authors need to properly discuss the relevant literature (see comment 1 below for details), since there is an important paper they didn't cite that already demonstrated some of the physics that the authors describe as new in the current manuscript. I also ask the authors to consider my other (minor) comments below.
1. The manuscript needs to properly discuss its results in relation to the work of Wen et al., Physical Review Letters 123, 233502 (2019). There, two superconducting qubits are placed close to the end of a transmission line. This corresponds to a setup with two small atoms in front of a mirror. As discussed in Ref. [5], this setup is essentially equivalent to the setup of two nested giant atoms studied in the current manuscript. The work of Wen et al. demonstrated a large coherent coupling between the two atoms, observable since the relaxation rate of one of them went to zero. This is essentially the same as the results discussed in the last paragraph on page 4 in the current manuscript. As noted in the text there, this effect would not be possible for small atoms in an open waveguide. However, small atoms in an open waveguide can demonstrate both purely coherent and purely dissipative coupling (see, e.g., Ref. [34]), while the current manuscript sometimes makes it sound as if this is something new for giant atoms. 5. At the end of the same paragraph, the advantages of spin ensembles over superconducting qubits are discussed. However, not all of the things listed seem to be actual advantages. While the magnon mode frequency can be adjusted in situ through a bias magnetic field, so can the frequency of a superconducting qubit (see Ref. [8], for example). Even if the magnons have better coherence at room temperature, don't they need to be cooled down to millikelvin temperatures anyway for this experiment at GHz frequencies? Actually, I don't see any specification in the paper about what temperature the experiment was conducted at. This should be stated.
6. In Fig. 1b, it would be good to have a scale bar or something similar to show what the length scales in the experiment are. 7. Below Eq. (4), when stating all the fitted parameter values, it would be good to also give the value of v, which now only was given at the end of the supplementary material.
8. Some typos to correct: * Page 1, second column, end of second paragraph: hybird --> hybrid * In several places: times of $\pi$ --> multiples of $\pi$ * Several places in the supplementary material: lamb shift --> Lamb shift Reviewer #2: Remarks to the Author: The manuscript entitled 'Giant spin ensembles in waveguide magnonics' from Zi-Qi Wang et al. reports on their experimental observations of the interactions between two spatially separated YIG spheres, i.e., the giant spin ensembles termed by the authors, meditated by a meandering waveguide.
The authors develop a method that utilizes the magnetocrystalline anisotropy to locally tune one YIG's resonant frequency. Then, they design a meandering waveguide that can interact with each YIG twice to enable the self-interreference of each YIG and the mediated coupling between two YIGs. They operate the microwave transmitting measurements at different frequencies to change the propagating phase delay between two YIGs. Consequently, they can control the selfinterference of each YIG and switch the coupling type, coherent or dissipative, between two YIGs. The paper is well written. The descriptions of both the experiment and the theoretical model are clear. The topic of the interactions between two resonant systems in a waveguide is important and of interest, because it may find applications in constructing quantum networks. Therefore, this topic attracts a lot of attention and has been studied in various physical systems, such as the superconducting qubit, acoustic systems and spin systems. However, I noticed that a similar work had been published in Physical Review Letters in this January [Phys. Rev. Lett. 128, 047701 (2022)], therefore I am nervous that this article written by Zi-Qi Wang et al. does not provide sufficient novel physics for publication in a high impact journal as Nature Communications.
In addition, I feel that some definitions and arguments in this manuscript may not be accurate. Please see the following comments: 1. By using a meandering waveguide, each YIG can interact with the waveguide twice. The authors call such YIG sphere the giant spin ensemble and emphasize the 'giant' physics. I'm not sure whether it is appropriate to rename a well-known system or not. I therefore would like the authors to provide further clarification.
2. The 'level attraction' shown in Fig. 4 (c) is so different from previous works [Phys. Rev. Lett. 128, 047701 (2022), or Ref.32]. Due to such difference, it seems that here in this work two resonances are only weakly coupled and so that only one resonance dip can be found near zero detunings. According to the authors' theoretical model, I agree that the dissipative coupling between two YIGs occurs. But it's too weak to produce a level attraction. Furthermore, the calculated results (dashed lines in Fig. 4 (c)) seem to be inconsistent with two modes' positions from experiment as they approach each other.
3. The authors state that they observe an unexpected coherent coupling between two YIGs, even when the outer YIG is decoupled from the waveguide. I feel it's a little bit wired. If that, the radiative damping rate of the outer YIG, i.e., \kappa_o, is zero. According to Eq. (7), the coherent interaction strength is zero, which means that the coherent coupling doesn't exist. This paradox needs to be further discussed in their manuscript.
4. The coupling between two YIGs arises from their radiative damping rates into the waveguide. Considering that both YIGs have Gilbert damping rates, it is very hard to tune this coupled system into a strong coupling regime. I notice that the authors give a strong criterion at the end of the discussion of the experiment. It may need to be clearly proved.
I believe that this work presents some interesting ideas. But with the previous comments in mind, the above questions need to be addressed to convince the readers that a sufficient level of progress/novelty was achieved for publication in Nature Communications.
Reviewer #3: Remarks to the Author: The authors have investigated 'giant' physics by exploiting ferromagnetic resonance modes of yttrium iron garnet (YIG) at room temperature in a meandering waveguide. The subject is quite interesting in the broad domain of light-matter interactions as well as magnonics. The authors have conducted detailed and precise experiments for the demonstrations. The data are supported by theoretical models. The figures are well-organized. The conclusions are supported by the results.
The novelty of the manuscript includes a new platform to study light-matter interaction at room temperature. Furthermore, this architecture offers ease of tunability/control as a ferromagnetic spin ensemble is used. It will potentially attract wide attention across domains. Therefore, I feel that this work is suitable for publication in Nature Communications. Nevertheless, the authors may want to address the following suggestions.
•It will be good for the readers to explicitly describe the sample details (YIG fabrication details, size etc.). •Similarly, include the waveguide details (fabrication details, size etc.). •Please comment on the choice for the cavity separation (L). Also, discuss the dependence of the waveguide design on the results. •The author may want to add a discussion on how to improve the effect based on their experience for future advancement. •It will be good to include an experimental image of the sample along with the meandering waveguide.
•Observation of anti-crossing of magnon modes has been observed in the past for different contexts. Can the author comment on such observations and distinguish their results? •Include the experimental details such as the power used for the RF current.
•Bias current has been used for the plots throughout the manuscript and it is misleading as it is used to drive an electromagnet. It is the bias field that is more appropriate. The authors may revise the relevant figures accordingly. •The Kittel mode has been investigated in this report for the demonstrations. However, it will be good to add a discussion on the variation of the higher order mode between the coupling state and the un-coupled state. •Comment on how to miniaturize such a sample design. •Add a discussion by comparing the results with previous demonstrations (superconducting qubit) that are already cited in the manuscript.
First of all, we would like to thank referee #1 for his/her quick, positive and constructive comments. Below we respond to them one by one.

Comment：
In this manuscript, the authors report the first experimental implementation of "giant atoms" using spin ensembles coupled to a waveguide. The usual situation in quantum optics is that atoms are much smaller than the wavelength of the light they interact with. It's only in the past decade that the possibility of (artificial) atoms interacting with light at multiple points, spaced wavelength distances apart, has been considered. There have been experimental demonstrations of such systems with superconducting qubits coupled to either microwave photons in meandering waveguides or surface acoustic waves, and proposals for some other implementations, but the present manuscript is to the best of my knowledge both the first proposal and the first experimental implementation with magnons in spin ensembles.

Reply:
We fully agree with referee #1's description of our manuscript and also appreciate his/her review on the state of the art of the giant atom physics.

Comment:
The experiments performed by the authors demonstrate thoroughly and convincingly how they can harness the interference effects due to the multiple coupling points of giant atoms. By tuning the resonance frequency of a spin ensemble using a bias magnetic field, the phase acquired when travelling between coupling points changes. This leads to either constructive or destructive interference, resulting in either high or low relaxation rate from the atom into the waveguide. The authors further demonstrate what happens when two giant atoms in a "nested" configuration (the two coupling points of one atom placed in between the two coupling points of the other atom) have their frequencies tuned. They find points where the coupling between the two atoms is purely dissipative and where it is purely coherent. In the latter case, the individual relaxation rate of one of the giant atoms also goes to zero.

Reply:
We thank referee #1 for this positive comment and agree with his/her comprehensive summary of our work.
The work of Wen et al. demonstrates the collective behavior of two superconducting qubits (small atoms) placed close to the end of a transmission line. In this study, the experimental results and effective Hamiltonian of the system are equivalent to the nested giant atoms. In the revised manuscript, we have appropriately cited this fascinating work (see new Ref. [33] cited in the revised manuscript). Correspondingly, we have added a sentence in the right column of page 1 as follows: "This phenomenon was previously observed with two superconducting qubits (small atoms) placed close to the end of a transmission line.".
It should be noted that in the two-port waveguide setup, the situation is different. As demonstrated by Van Loo et al. (i.e.,Ref. [38] in the revised manuscript), the interaction between two superconducting qubits (small atoms) is mediated by the standard two-port waveguide. The coupling (either coherent or dissipative) between these two small atoms can be regulated by the resonance frequency, but if one of the two small atoms is decoupled from the waveguide, there is no longer any interaction between these two small atoms. However, a novel phenomenon can occur in the case of giant atoms. Our work highlights the counter-intuitive phenomenon that even the outer giant spin ensemble is decoupled from the two-port waveguide, we can still observe the coherent interaction between the two giant spin ensembles. This phenomenon is incapable with two small atoms in the traditional two-port waveguide configuration, as established in Ref. [38].
Nevertheless, referring to the work by Wen et al. (Ref. [33]), it is indeed possible to construct coherent interaction between small atoms when one of the atoms is decoupled from the environment in the configuration of transmission line with an end. In order to avoid any misunderstanding, we have thoroughly revised the text relevant to all potentially misleading descriptions. For example, on page 5, we have added the phrase "two-port waveguide" while discussing and comparing with the waveguide-mediated interaction between small atoms studied by Van Loo et al. (Ref. [38]). Also, we have added an explanation in the last paragraph of page 1 as follows: "In contrast, the coupled giant spin ensembles are implemented in the two-port waveguide system".
In addition, we would like to underline that the waveguide configuration associated with 'giant atom' physics may have applications in the construction of a quantum network, as it is based on a two-port setup that extends in both two directions, in contrast to the transmission line with an end.

Comment:
2. In the introduction, the expression "giant physics" is used several times. I ask the authors to find a better wording in those places. Since nothing is said about atoms, light, or quantum optics in that expression, it could be misunderstood. For example, cosmology deals with giant length and time scales. Isn't that also "giant physics" then?

Reply:
We thank referee #1 for this helpful comment. In the revised manuscript, Ref.

Comment:
5. At the end of the same paragraph, the advantages of spin ensembles over superconducting qubits are discussed. However, not all of the things listed seem to be actual advantages. While the magnon mode frequency can be adjusted in situ through a bias magnetic field, so can the frequency of a superconducting qubit (see Ref. [8], for example). Even if the magnons have better coherence at room temperature, don't they need to be cooled down to millikelvin temperatures anyway for this experiment at GHz frequencies? Actually, I don't see any specification in the paper about what temperature the experiment was conducted at. This should be stated.

Reply:
We thank referee #1 for this thoughtful question. Compared with the superconducting qubits, the advantage of the flexibility and adjustability of the magnon mode is mainly manifested in its large continuously tunable frequency range. The magnon mode frequency can be tuned from a few hundred megahertz to several tens of gigahertz by manipulate the bias magnetic field, which is hard to realize with superconducting qubits. Our work also utilizes the magnetocrystalline anisotropy field to tune the frequency of magnon mode by rotating the YIG sphere in a static magnetic field, which is favorable for individually tuning the magnon mode frequency in different YIG spheres. However, in order to avoid overemphasizing the advantages of spin ensembles while ignoring the unique advantages of superconducting qubits in demonstrating the physics of giant atoms, we have deleted "rather than a superconducting qubit". Correspondingly, we have modified the description of the advantage as follows: "The benefits of choosing a ferromagnetic spin ensemble include the flexible and wide-range adjustability of magnon mode frequency (a few hundred megahertz to several tens of gigahertz), …".
Actually, our experiment was conducted at room temperature. The expression "favorable magnon coherence at room temperature" in this paragraph is intended to illustrate the low dissipation rate of the magnon mode at room temperature. Since the Curie temperature of yttrium iron garnet is as high as 550 K, the ferromagnetic spin ensemble can be used to demonstrate the self-interference effect related to the giant atom physics at room temperature. To make the statement clear, we have revised the expression as follows: "...low dissipation rate at room temperature..." and added the specification of the experimental temperature in the right column on page 2.

Comment:
6. In Fig. 1b After carefully reading the PRL work, we see the difference and disparity between our research goal and that of the PRL work. The objective of this PRL work is to investigate the long-range interactions between spin ensembles mediated by superconducting circuits, which does not involve the specific structure and physics of giant atoms. However, our primary objective is to illustrate the 'giant atom' physics in spin ensemble systems, exploiting the rich interference effect of 'giant atom' physics to manipulate the interaction between the spin ensemble and its environment, as well as the interaction between spin ensembles. Our central results involve a new light-matter interaction beyond the dipole approximation, which is not studied in the PRL work above. In this regard, we are convinced that our experimental demonstrations are sufficiently novel. Below we would like to describe in detail the primary novel aspects of our work. We sincerely hope that referee #2 can find these explanations useful to see the novelty of our work.
In the small spin ensemble case, the YIG sphere is treated as a point in the light-matter interaction. For example, in the PRL work of Yi Li et al., the size of the YIG sphere (~250 μm) is much smaller than the wavelength of the microwave field (on the order of a centimeter) with which it interacts, as shown in Figure R1a below. However, in our work, the size of the YIG sphere is also small, but its effective size is effectively enlarged by using the meandering waveguide, which is also on the order of a centimeter. This non-negligible effective size of the YIG sphere (~cm) in the light-matter interaction leads to the selfinterference effect between the light-matter coupling points. This effect is well reflected in the frequency-dependent relaxation rate of the magnon mode in the YIG sphere, and the periodic decoupling of the single spin ensemble from the environment (waveguide) is observed, as depicted in Figs. 2a and 2b of the main text. This is an essential feature of 'giant atom' physics that cannot be realized with small atoms and small spin ensembles. This enables us to realize the switching between decoupling and coupling by simply adjusting the frequency.
We further demonstrate the intriguing collective behavior of two nested giant spin ensembles. In the coupled giant atoms system, the different topologies of the coupling points can give rise to various interaction behaviors. A representative example is the decoherence-free interaction in braided structure depicted in Fig. R1c, which is examined experimentally using superconducting qubits [Kannan et al., Nature 583, 775-779 (2020)]. As depicted in Fig. R1b, our work investigates experimentally for the first time the nested configuration. In this setup, we find that the coupling between giant spin ensembles mediated by meandering waveguide and the coupling between small spin ensembles mediated by conventional waveguide [as demonstrated in Yi Li et al., Phys. Rev. Lett. 128, 047701 (2022)] are very different. Even when the outer giant spin ensemble is decoupled from the waveguide, a level repulsion is observed. In contrast, in the small spin ensemble system, when one of the spin ensembles is decoupled from the waveguide, the interaction between the spin ensembles becomes unavailable. This novel phenomenon in our system of giant spin ensembles can stimulate further investigations. Moreover, in the following responses to specific questions, we would like to explain in detail that in the nested giant atom system, it is feasible to achieve strong coherent interactions through the mediation of open waveguides. However, in small atom systems, strong coherent coupling mediated by a waveguide is unachievable. This unique effect will enable one to control the interactions between nodes in future quantum networks in a flexible and efficient manner. Finally, we would like to mention that, to the best of our knowledge, this is the first experimental realization of the nested structure of giant atoms/spin ensembles. Since the rapid theoretical development of the 'giant atom' physics (see  in the main text), only superconducting circuits have been employed to demonstrate novel light-matter interaction beyond the dipole approximation. Our demonstration has timely provided a new and easy-tooperate platform to investigate the versatile effects of 'giant atom' physics. This novelty is acknowledged by referee #1's comment: "…but the present manuscript is to the best of my knowledge both the first proposal and the first experimental implementation with magnons in spin ensembles." Through the detailed explanations above, we hope that referee #2 finds our work to be original in exploring the physics of 'giant atoms' by using unique configurations of spin ensembles interacting with a waveguide.

Comment:
In addition, I feel that some definitions and arguments in this manuscript may not be accurate. Please see the following comments:

Reply:
We thank referee #2 for helpful and constructive comments. We have studied the rest of referee #2's comments and addressed them accordingly. Below we respond to them one by one.

Comment:
1.By using a meandering waveguide, each YIG can interact with the waveguide twice. The authors call such YIG sphere the giant spin ensemble and emphasize the 'giant' physics. I'm not sure whether it is appropriate to rename a well-known system or not. I therefore would like the authors to provide further clarification.

Reply:
We thank referee #2 for this helpful comment. According to the original Kockum's study (Ref. [3] in the main text), when the size of the atoms becomes comparable to the wavelength of the field with which they interact, the dipole approximation no longer applies and the atom is referred to as a 'giant atom'. The giant atom leads to a series of striking effects incapable with the small atoms (Refs. [17][18][19][20][21][22]). The non-local interaction between the atom and the field is the basis of 'giant atom' physics (depicted in Fig. R2). We use the meandering waveguide to have the spin ensemble interact several times with the field, and this scheme is purposed by Kockum (Ref. [3]) and implemented for the first time by Kannan in a superconducting circuit (Ref. [5]). This concept prompted us to explore the physics of 'giant atoms' on a novel platform using YIG spheres. The structure of the meandering waveguide enables the traveling photons interact twice with the magnon mode in the spin ensemble, resulting in the effective enlargement of the spin ensembles. As also noted by referee #1, the expression 'giant physics' is certainly misleading. Therefore, we have modified the 'giant physics' to 'giant atom' physics in the revised manuscript. Figure R2. Illustrations of small atom (left), and giant atom (right).

Comment:
2. The 'level attraction' shown in Fig. 4 (c) is so different from previous works [Phys. Rev. Lett. 128, 047701 (2022), or Ref.32]. Due to such difference, it seems that here in this work two resonances are only weakly coupled and so that only one resonance dip can be found near zero detunings. According to the authors' theoretical model, I agree that the dissipative coupling between two YIGs occurs. But it's too weak to produce a level attraction.

Reply:
We thank referee #2 for this insightful comment. Because of the difference in relative magnitude between the strength of the dissipative coupling and the dissipation rates of the coupled modes, the 'level attraction' depicted in Figure 4c appears different from the previous work. In other words, this is because they are in different dissipative coupling regimes.
First, let us briefly review the various coherent-coupling regimes. According to the definition in the work of X. Zhang et al., Phys. Rev. Lett. 113, 156401 (2014), in a two-mode coherently coupled system (A mode and B mode), by comparing the coherent coupling strength ( ) and the dissipation rate of each mode (κ , κ ), the coupled system can be divided into weak coupling region ( < {κ , κ }), strong coupling region ( > κ , κ ), magnetically induced transparency (MIT) region ( κ < < κ ), and Purcell effect region ( κ < < κ ). In different coupling regions, different spectral mappings can be obtained, as shown in Fig. R3 (c) and (e) [adapted from Phys. Rev. Lett. 113, 156401 (2014)] corresponding to the magnetically induced transparency regime and Purcell effect regime, respectively. Despite the diverse appearances of the spectral mappings, the spectral features are always characterized by energy level repulsion, which is the manifestation of coherent coupling. Figure R3. Different coupling regimes of coherently coupled system. (c) Spectral mapping obtained in the magnetically induced transparency regime; (d) Spectral mapping obtained in the Purcell effect regime [adopted from Phys. Rev. Lett. 113, 156401 (2014)].
It is shown in Fig. R4a that the 'level attraction' is induced by the dissipative coupling between two resonators, in which the resonators cooperatively dissipate into the open environment. Similarly, regimes of dissipative coupling in a two-mode system can be classified. The magnitude of the dissipative coupling strength is equal to the square root of the product of two external dissipation rates of the resonators = √ κγ, and each resonator has its own intrinsic dissipation rate (α and β). Therefore, the quantities to be compared are the strength of the dissipative coupling = √ κγ and the total dissipation rates of the two resonators κ + α and γ + β.
Obviously, in a dissipative coupling system, the dissipative coupling strength cannot exceed the total dissipation rates of the two resonators at the same time ( = √ κγ > {κ + α, γ + β}), i.e., the dissipative coupling system cannot reach the strong coupling regime. However, there are circumstances in which the external dissipation rate of one of the resonators is significantly greater than the external dissipation rate of the other resonator and the intrinsic dissipation rates of the two resonators are relatively small. In this case, the magnetically induced transparency regime (κ + α > √ κγ > γ + β) and Purcell effect regime (γ + β > √ κγ > κ + α) can be achieved. In more cases, one can only achieve weak coupling regime of dissipative coupling, that is, when the two resonators are relatively equivalent (κ ≈ γ, α ≈ β, √ κγ < {κ + α, γ + β}).
As illustrated in Fig. R4b, we distinguish the different dissipative coupling regimes by comparing the relative magnitudes of the dissipative coupling strength and the damping rates of two resonators. The previous work [M. Harder et al., Phys. Rev. Lett. 121, 137203 (2018)] belongs to the magnetically induced transparency (MIT) regime, and our work is in the weak coupling regime. In the work of M. Harder et al., the 'level attraction' occurs between the cavity mode and the magnon mode, where the cavity mode's external damping rate is two orders of magnitude greater than the magnon mode's external damping rate, so the system is in the MIT regime. Using the same parameters in M. Harder's PRL work, the transmission mapping can be calculated and plotted in Fig. R4c. It shows the typical 'level attraction' appearance. The MIT window appears and passes through the broad cavity resonance. The transmission spectrum at resonance is shown in Fig. R4e and an MIT peak is clearly visible. The result is consistent with the experimental result demonstrated in the PRL work of M. Harder. Figure R4. a. Schematic of the dissipative coupling mechanism; b. Different coupling regimes of the dissipatively coupled resonators; c. Level attraction in the magnetic induced transparency regime of the dissipative coupling; d. Level attraction-like effect in the weak coupling regime of the dissipative coupling; e. and f. Transmission spectra corresponding to the resonance positions in c and d, respectively.
In our work, dissipative coupling occurs between two giant spin ensembles with comparable damping rates, indicating that the system is in the weak coupling regime. Utilizing the same parameters as in our work, the transmission mapping is plotted in Fig. R4d. We can find that two modes are degenerate at resonance, and the transmission spectrum exhibits a Lorentzian broadened line shape, as depicted in Fig. R4f. Although it is not as typical as the instance observed in the MIT regime, the spectral energy level attraction also reflects the dissipative coupling. In order to distinguish the energy level attractions observed in the previous work and in our present work, we modified the phrase 'level attraction' as 'level attraction-like' in the revised manuscript.
In addition, as indicated by referee #2, the level attraction observed in the other paper [Phys. Rev. Lett. 128, 047701 (2022)] also appears distinct from ours. We think that the situation in this PRL work is also essentially in the weak coupling regime, since the dissipative interaction is demonstrated between two nearly identical YIG spheres. Based on the transmission mapping, we infer that the difference may stem from higher-order magnetostatic modes that are involved in the dissipative coupling. Near the resonance, the spectral lines of other magnetostatic modes can be seen in this PRL work. However, only Kittel modes are involved in our work, without contributions from the higher-order magnetostatic modes.
Since the strong coupling region of dissipative coupling cannot be reached with the current technology, and the magnetically induced transparency regime has been commonly referred to as energy level attraction, we refer to the spectral lines with the attractive characteristics of dissipative coupling in other regimes as energy level attraction-like. Based on referee #2's suggestion, we have modified the manuscript accordingly.

Comment:
Furthermore, the calculated results (dashed lines in Fig. 4 (c)) seem to be inconsistent with two modes' positions from experiment as they approach each other.

Reply:
We thank referee #2 for the helpful comment. According to the referee suggestion, we have revised the calculated result (dashed line) in Fig. 4c of the main text.

Comment:
3. The authors state that they observe an unexpected coherent coupling between two YIGs, even when the outer YIG is decoupled from the waveguide. I feel it's a little bit wired. If that, the radiative damping rate of the outer YIG, i.e., \kappa_o, is zero. According to Eq. (7), the coherent interaction strength is zero, which means that the coherent coupling doesn't exist. This paradox needs to be further discussed in their manuscript.

Reply:
We thank referee #2 for this helpful comment. Equations (4) and (7) in the main text describe the phase dependence of the radiative decay rate and the coherent coupling strength, respectively. It should be pointed that the is defined as the radiative damping rate of the outer giant spin ensemble at the coupling points, which is fitted to be 0.70 MHz. Equation (4) gives the radiative damping rate of the giant spin ensemble , , which is zero in the case of purely coherent coupling, where the subscript G denotes 'giant'.
Intuitively, we think that the two giant spin ensembles will not interact with each other since the outer giant spin ensemble is decoupled from the waveguide. Nevertheless, Eq. (7) and Fig. 3b reveal a non-zero coherent coupling between two giant spin ensembles, which is clearly shown in Fig. 4g. This unexpected novel phenomenon is explained in the right column of the main text on page 4.
We recognize that the subscripts of and , may lead to some misunderstandings. In the revised manuscript, we have elaborated on these parameters at the given places.

Comment:
4. The coupling between two YIGs arises from their radiative damping rates into the waveguide.
Considering that both YIGs have Gilbert damping rates, it is very hard to tune this coupled system into a strong coupling regime. I notice that the authors give a strong criterion at the end of the discussion of the experiment. It may need to be clearly proved.

Reply:
We thank referee #2 for this insightful comment. In our work, the damping rates of the nested two giant spin ensembles consists of the intrinsic damping rates , and radiative damping rates , , , . The Gilbert damping rate is included in the intrinsic damping rate. It should be noted that each giant spin ensemble is here designed to have two coupling points with the meandering waveguide. The radiative damping rates at the couplings points of the inner and outer giant spin ensembles are , respectively. According to Eq. (7), it is obvious that the maximal value of the coherent coupling strength is proportional to the product of the radiative damping rates of the inner and outer giant spin ensembles at the coupling points ( ∝ √ ). Despite the outer giant spin ensemble being decoupled from the waveguide (i.e., , = 0), the coherent coupling strength still exists, as mentioned above. Utilizing this advantage, if the radiative damping rate of the outer resonator at the coupling point is large enough (like the superconducting qubits), then we can have √ > { , + , }, and the system will enter the strong coupling regime in the open environment.

Comment:
I believe that this work presents some interesting ideas. But with the previous comments in mind, the above questions need to be addressed to convince the readers that a sufficient level of progress/novelty was achieved for publication in Nature Communications.

Reply:
We are very grateful to referee #2 for providing us with many valuable and insightful suggestions and comments. We fully adopted the comments and revised the unclear parts of the manuscript, so as to eliminate the misunderstandings. Also, we have carefully explained the important achievements and novelties of our manuscript regarding the demonstration of the intriguing 'giant atom' physics in the new platform of spin ensemble system. We sincerely hope that our answers can be satisfactory to referee #2.